It was another busy night at Hotel Infinity. I think it was about 8pm although it could have been a little later. The time wasn't really all that important. What was important was that word had just been received that the Queen of England would arrive shortly and expected a room for the night.

Unfortunately, every single one of the Hotel's infinite rooms was full.

Fortunately, the concierge, who had a Phd in Mathematics but had gotten tired of driving a taxi and had decided to get a real more discrete job, suggested that the duty manager move each of the guests from their current room to the next higher numbered room as this would make room 1 available without requiring that any guest be evicted. The duty manager replied that this would almost certainly result in a lot of complaints although the concierge was quick to point out that with an infinite number of guests, there were going to be an infinite number of complaints anyways.

That was good enough for the duty manager and the plan was put into operation. When the Queen arrived, she was presented with the keys to room 1 (the Royal Suite, of course) and everyone except the concierge was happy (the concierge had realized too late that although there would probably be a big tip from the Queen in the morning, there wasn't going to be the usual infinite number of tips from satisfied guests).

#### What's the point of this story?

This story is intended to illustrate a few of the more subtle aspects of the concept of infinity:
• that adding one (i.e. the Queen) to a set of infinite size (i.e. the number of guests already in the Hotel when the Queen arrives) doesn't change the size of the set

• that you can map (i.e. match up) the elements of one infinite set (i.e. the rooms in Hotel Infinity) with the elements of another infinite set (i.e. the guests staying at Hotel Infinity) and that adding a finite number of elements to one set (i.e. adding the Queen to the set of guests) doesn't affect our ability to map the elements (i.e. put guests into rooms without leaving any guests without a room).

There's a variation of this story which involves adding an infinite number of guests to the Hotel after it's full - this works also as doubling the size of an infinite set doesn't change the size of the set (i.e. infinity plus infinity is equal to infinity)

• that if there is a non-zero probability that each element of a set of infinite size will have a certain property (i.e. the non-zero probability that each guest will complain about something) then the number of elements of the set with the property (i.e. the guests that complain about something) is infinite

• that increasing the probability that an element of the set has the property (i.e. doing something that DRAMATICALLY increases the probability that a guest will complain) doesn't change the size of the set of elements with the property (i.e. size of the set of guests which complains)

• that a very small amount times infinity (i.e. the average size of a tip left by each guest times the infinite number of guests) is infinity which is (infinitely) larger than the largest possible finite value (i.e. if none of the guests except the Queen leaves a tip then the size of the tip will be finite which is much MUCH worse than the infinite tip that the concierge usually gets when the Hotel is full)
There's also a stupid joke surrounding the term "real" - Hotel Infinity has an infinite number of "discrete" rooms (i.e. like the infinite set of natural numbers and unlike the "larger" infinite set of "real" numbers - see the infinity node for an explanation of why there are more real numbers than integers).

On the other hand, I did (until now) resist the temptation to say that the concierge used to have a continuous job but now has a discrete job (sorry - no explanation of this even stupider "inside joke").

#### Source

A version of this story was told in class by my second year calculus professor. It was a key step along the way to my comprehending the notion of "infinity".

This story is often told with the hotel called the Hilbert Hotel. I prefer the name "Hotel Infinity" as the story flows better with that name and calling it the "Hilbert Hotel" would make it necessary to explain why it is called the "Hilbert Hotel" whereas the name "Hotel Infinity" is, I believe, pretty obvious.

An instant after agreeing to come aboard the Andromedan ship, we are aboard, standing in a plain white space which is presumably somewhere inside it. This comes as something as a shock since we were previously seated on the Ed Rocks.

"Now wait a second," I interject. "I can buy that the Andromedans have figured out how to tunnel objects between two different gravitational potentials. I can buy that the relative velocity of us with respect to this ship was most likely counteracted by a burst from a momentum cannon at the instant we arrived. But we were accelerating around in a fifty-kilometre circle to generate gravity, while the Andromedan ship was just in free-fall orbit around Epsilon Eridani One."

"So?"

"So how come there's gravity in here?"

"You said it wasn't possible to fake a graviton."

"It's not. Observe." Ed rummages in a pocket and pulls out a screwdriver and a set of keys - neither of which appear to be suitable for what he has in mind. "Ship, can I have a rubber ball please? And can you also switch off gravity for it?"

From nowhere, a small red ball appears in Ed's hand. Ed puts the ball out in mid-air and lets it go. It hangs there, spinning, while we stand normally on the ground.

"Whoo," I remark, impressed.

"Now watch this. Ship, give the ball a small downward impulse."

The ball begins to drop slowly.

"And another."

The ball drops more quickly.

"And another."

The ball falls and hits the ground, where it rebounds.

"So the ball fell," says Ed as the ball rises. "Now let's do it again, only giving it half the impulse, twice as often."

The ball rises, slowing in stages until it's almost not rising at all, then it's falling slightly, then it drops in stages until it rebounds again. It's almost a piece-by-piece approximation of a proper arc.

"Again. Make it finer still."

This time I can hardly tell that the ball isn't following a proper parabola.

"Now give it a standard one gravity."

The ball bounces just like it was on Earth.

"Do you get it? Force is just continuous change in momentum. If you tune your cannon finely enough, nobody can tell the difference between constant gravity and just being knocked downwards very slightly every few nanoseconds. No fake gravitons, no mucking about with neutronium. The best part is, you can have any number of people walking around in the same room, but experiencing whatever gravity is comfortable for them. A guy built for one-sixth of a gee can meet and shake hands with a guy built for twenty-five. You can have people walking on walls or the ceiling with consummate ease. All you need is a sufficiently smart computer to keep an eye on everything."

"May we talk now?" asks the ship. It addresses us from behind, using a deep male voice with no obvious accent. We turn around to see what, for the Andromedans, would probably count as a very crude humanoid robot. "I'm here to serve as a focal point for our conversation," it explains. Three large armchairs materialise around us. "Do be seated."

We take one each. For an instant I feel intensely worried. The Andromedans are powerful, I always knew that, but I obviously never fully registered how powerful. This ship, for example, can pull you across thousands of kilometres of space in an instant and leave behind, say, your watch. And it can manufacture furniture on-the-fly. And robots. Doubtless the one we're talking to could easily have been indistinguishable from a human being; the only reason it isn't is because that would be off-putting.

Thank heaven they're not hostile.

"First, let me introduce myself," says the robot. "I am the Xenosociology Research Vessel Hotel Infinity. I am Andromedan, specifically a citizen of a conglomerate of seven and a half civilisations occupying a small portion of one of Andromeda's lesser spiral arms."

"Seven and a half civilisations?" I ask.

"That would be too complicated to go into now. My civilisation and species is Iisium; I am an Iisius. We might be described as living ships, although only an average of five percent of a given Iisius's body is what one would describe as 'living', the remainder being hard mineral shell built up over hundreds of years of growth. We evolved in the inner asteroid belt of a very young star. Interplanetary travel is as natural to us as walking is to you, though more complicated machines such as tunnel drives, momentum cannons and nanoassemblers are not evolutionary characteristics of ours.

"Iisii typically live for two to four thousand years, and we are relatively patient as species go. Beyond that, it's difficult to be specific as we, like all species, are extremely diverse. I personally am a freelance xenosociologist. I don't usually carry passengers as I find maintaining habitat for them is tiresome and they interfere with my work, but this is perfectly normal behaviour for an Iisius. As I say, we vary.

"You needn't introduce yourselves; I have already been told who you both are. This will become clear.

"Time travel is an extremely well-understood science in Andromeda. The theory is long established and the principles behind its practice likewise. However, attempting to travel backwards in time is illegal and punishable by rider. The reasons for this are twofold.

"Firstly, prescience is very, very powerful and dangerous. Being able to guess what the future holds is one thing. Knowing with certainty because it already happened is quite another. An individual with knowledge of the future could use that knowledge to do literally anything - he is only limited by his imagination. He could rule his world. He could rule his galaxy. He could reshape the universe in his image. Nobody can be allowed to have that much power.

"Secondly, backward travel does not undo the past. Rather, it causes the past to repeat. If you go back in time with the intention of undoing some cataclysmic event - say, a genocide - then you have failed - everything which happens stays happened. What you have done, rather, is give everybody involved in that genocide a chance to die again. If ten million people died, and you go back and make it so only one million people died, you haven't saved nine million lives. You've killed a million people.

"For these reasons, it is as difficult for a typical Andromedan citizen to travel backwards in time as it is for a Human to, say, obtain a kilogram of weapons-grade plutonium. Nobody is allowed to go back, regardless of the circumstances, regardless of the catastrophe they are hoping to undo. However, this is not to say that nobody does go back. It is, after all, a big universe. There are other civilisations out there discovering things all the time.

"There are scientific stations all over the galaxy which can detect and log the shifts which are caused when someone departs our universe and heads backwards in time. Also detectable are the shifts when somebody arrives in our universe after having headed backwards in time. But these are far rarer. Only two of these have ever occurred. That's two more than all probability predicts.

"The first shift happened sixteen years ago. Unfortunately, while the bifurcation is detectable, the location of the incursion is not. We had no way to find out who had come back in time in order to create our universe. This has been one of science's greatest unsolved mysteries ever since.

"The second shift happened yesterday, and was equally inexplicable... for a few hours. Then the Raft Mantissa and its occupants returned to Andromeda. Krah's story was told. And while most people's minds are still focused on the act of galacticide which occurred at Ed's hand in that alternate timeline, and his attempt to make amends for that terrible mistake, those Andromedans such as myself who take an interest in intercivilisational wars, first contact and so on, looked at the deeper details about your world, your history, and you specifically, and discovered that Krah's story may in fact explain both shifts. I'm not saying anything yet, as I have no proof of my hypotheses. But I believe I know where I can find it. Ed, may I have permission to interface with your mind? The rider you are already carrying should make the process swift and painless."

Ed glances nervously at me, then, with hesitation, nods his consent. The robot gestures, more likely for show than anything.

And Ed begins to recite, apparently from memory—

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